Objective Functionals of Machine Learning on Graphs and their Continuum Limits

We will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying ground-truth measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and semi-supervised learning, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts, graph laplacians and their limits as the number of data points goes to infinity. We will discuss the limits of functionals when the number of data points goes to infinity. In particular we establish under what conditions the minimizers of discrete problems have a well defined continuum limit.